Centralizer - Knowlpedia

Let $G$ be a group and $S\subseteq G$ a subset. The centralizer of $S$ in $G$ is the subgroup

C_G(S)=\{g\in G:gs=sg\text{ for every }s\in S\}.

For $x\in G$ , one writes $C_G(x)$ for $C_G(\{x\})$ .

Remarks

For the conjugation action, $C_G(x)$ is the stabilizer of $x$ , so it controls the size of the conjugacy class of $x$ . The center satisfies $Z(G)=C_G(G)$ .

Examples

If $G$ is abelian, then $C_G(S)=G$ for every $S\subseteq G$ .
In $S_3$ , $C_{S_3}((12))=\{e,(12)\}$ .
In $S_3$ , $C_{S_3}((123))=\{e,(123),(132)\}$ .